mathwikiaorg-20200223-history
Wedge product
The Wedge product is the multiplication operation in exterior algebra. The wedge product is always antisymmetric, associative, and anti-commutative. The result of the wedge product is known as a bivector; in \R^3 (that is, three dimensions) it is a 2-form. For two vectors u''' and '''v in \R^3 , the wedge product is defined as : \mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = \begin{bmatrix} 0 & u_1 v_2 - u_2 v_1 & u_1 v_3 - u_3 v_1 \\ u_2 v_1 - u_1 v_2 & 0 & u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 & u_3 v_2 - u_2 v_3 & 0 \end{bmatrix} where ⊗ denotes the outer product. Note that the bivector has only three indepedent elements; as such, it can be associated with another vector in \R^3 . If the associated vector is defined as : \mathbf{n}= \begin{bmatrix} (\mathbf{u} \wedge \mathbf{v} )_{23} \\ -(\mathbf{u} \wedge \mathbf{v} )_{13} \\ (\mathbf{u} \wedge \mathbf{v} )_{32} \end{bmatrix} = \mathbf{u} \times \mathbf{v} it is the same as the cross product of u''' and '''v. In this sense, the cross product is a special case of the exterior product. Cross product and wedge product when written as determinants are calculated in the same way: : \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix} \,,\quad \mathbf{a} \wedge \mathbf{b} = \begin{vmatrix} \mathbf{e}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix}\ , so are related by the Hodge dual: : {* (\mathbf a \wedge \mathbf b )} = \mathbf {a \times b} \,,\quad {* (\mathbf {a \times b} )} = \mathbf a \wedge \mathbf b . In the language of Geometric algebra \mathbf{u} \wedge \mathbf{v} is the dual (orthogonal complement) of \mathbf{u} \times \mathbf{v} : : \mathbf{u} \wedge \mathbf{v}= (\mathbf{u} \times \mathbf{v})I Bivectors are skew-symmetric matrices which are the type of matrices used to calculate the cross product. Bivectors are not rotation matrices but an infinitesimal bivector (plus the identity matrix) can be used to perform an infinitesimal rotation. \mathbf{n} would therefore be the axis of rotation. See Rotation_matrix#Determining_the_axis. Cross product :See also:Cross_product#Conversion_to_matrix_multiplication The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: : \mathbf{a} \times \mathbf{b} = \mathbf{a}_{\times} \mathbf{b} where [a]× is defined by: : \mathbf{a}_{\times} \stackrel{\rm def}{=} \begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}. \mathbf{e_1}_{\times} = \mathbf{e_3} \wedge \mathbf{e_2} = \left\begin{matrix}0&0&0\\0&0&-1\\0&1&0\end{matrix}\right \mathbf{e_2}_{\times} = \mathbf{e_1} \wedge \mathbf{e_3} = \left\begin{matrix}0&0&1\\0&0&0\\-1&0&0\end{matrix}\right \mathbf{e_3}_{\times} = \mathbf{e_2} \wedge \mathbf{e_1} = \left\begin{matrix}0&-1&0\\1&0&0\\0&0&0\end{matrix}\right This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors. References See also *Exterior algebra *Exterior derivative *Differential form *Skew-symmetric matrices can be thought of as infinitesimal rotations *Rotation_group_SO(3)#Infinitesimal_rotations *Regressive product This article incorporates text from Wikipedia:Cross_product Category:Differential forms